Infrared gluons in the stochastic

quantization approach

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Contents1. Introduction2. Method: Stochastic gauge fixing3. Gluon propagators4. Numerical results5. Summary

Takuya Saito （ Kochi), Nakagawa Yoshiyuki 　(Osaka),

Nakamura Atsushi 　 (Hiroshima), Toki Hiroshi 　 (Osaka)

Introduction(1)

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ConfinementQuarks and gluons are basic quantities of QCD. In ultraviolet region, the perturbative QCD works well but in the confining region, some non-perturbative modes dominates hadron physics.Infrared physics of QCD: Confinement, Chiral symmetry breaking; these non-perturbative phenomena are deeply related to infrared singularities of QCD.

Infrared (transverse) gluon propagatorsIf confinement exists, one can expects that a transverse gluon propagator has an infinite mass, and will vanish in the IR limit. On the other hands, the ghost propagator diverges in the IR limit. We can find many lattice studies for these in many references; however, there are no distinctive signals, particularly for gluons.

Introduction(2)

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Numerical difficulty :

Finite volume size effect; the infrared physics requires large lattices.Gauge fixing computation on the large lattices is very hard, time-consuming simulations if we use the iterative gauge fixing.

Conceptual difficulty:Lattice configuration can not be gauge-fixed uniquely due to Gribov ambiguity. We expect that the Gribov copy configuration will fade the infrared physics we are interested in.Gribov copy problem is not fully understood now.

=== Some difficulties for lattice calculations for gluons ===

Introduction(3)

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Calculations of the gluon propagator in the stochastic quantization with the Coulomb gauge

This method has some advantage:We do not use the iterative gauge fixing method.Gauge configurations go to the Gribov region automatically.Gauge parameter is easy to change.

Measure of the transverse gluon propagatorsTransverse gluon propagator is a physical quantity.We expect that the gluon propagator in the infrared limit will be suppressed with an infinite effective masses. This means gluons are confining.

=== Aim in this study ===

Method(1)

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=== Stochastic quantization with the gauge fixing ===

Stochastic Gauge fixing ：D.Zwanziger,Nucl.Phys.B192(1981)

ababa

a

AADAS

ddA

)(1

Langevin equation for the gauge theory with the gauge fixing ( a la Zwanziger)

: Virtual time for the hypothetical stochastic process

::

Gauge parameter

Gaussian white noise

0),( x )()(2),(),( '''' xxxx

Method(2)

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=== Stochastic quantization on the lattice ===

),ˆ(),()exp(),(),( xxUtifxxU aa

a

aa

ASf

exp(iat a / )

Lattice generalization of stochastic gauge fixing ： A.Nakamura and M. Mizutani, Vistas in Astronomy (Pergamon Press,1993), vol.37 p.305. , M. Mizutani and A.Nakamura, Nucl. Phys. B (Proc.Suppl.)34(1994),253.

Driving force

Gauge rotation

Method(3)

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=== Conceptual reason for using SGF ===

Conceptual reasonGauge copy problem

Gauge configurations not fixed completely on the non-perturbative lattice calculationGauge fixing term of SGF 1. It makes gauge

configurations go to the Gribov region.

2. This term works as an attractive driving force.

3. More effective approachGribov region

0..,0)( PFxA

)(xA

Method(4)

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=== Practical reason for using SGF ===

Practical reasonFor a gauge fixing, we don’t use any iterative methods and so there is no critical slowing down of this algorithm. It is a great advantage for large lattice simulation with gauge fixing.Changing a gauge parameter is easier than the iterative methods.

AMonte Carlo Steps

～ Monte Carlo Quantization ～

Gauge rotations

～ Stochastic Quantization ～ A

Langevin steps

Coulomb gauge QCD

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=== basic issues ===

Hamiltonian of Coulomb gauge QCD

A transverse part makes a physics gluon field.

A source term makes a color-Coulomb instantaneous (confining ) potential among quarks, causing by a singular eigenvalue of F.P.

No negative norm : A physical interpretation is very clear.

3 2 2 3 31 1 ( ) ( , ) ( )2 2i iH d x E B d xd y x D x y y

Gluon propagators(1)

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=== General form in the perturbative region ===

General form of gluon propagators

For free case, we have

If adding an anomalous dimension, we have

)()()( 22

22 qD

qqq

qDq

qqgqD Lab

22

01)(q

qD

22 1~)(

qqD

Gluon propagators(2)

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=== Assumptions in the non-perturbative region ===

Mandlestam hypothesise ( if the confining potential is linear )

Gluon propagator with an effective mass

Gluon propagator vanishes in the IR limit

0 ,1~)( 24

2 qq

qD

222 1~)(

mqqD

44

22 ~)(

mqqqD

) Zwanziger todue ( 0)0(lim 2

qDN

Gluon propagators(3)

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=== Gluon propagators on the lattice ===

Gauge field on the lattice in this calculation

Fourier transform

Gluon correlators ( we’ll measure )

aa xUxUi

xA )()(Tr21)(

dxxAepA axipa )()2(

1)( 4

)()()( pApAqD baab

Numerical parameters

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β α Δτ L Thermal. Confs.

6.0 1.0 0.01 12 10k 100 (100)

6.0 1.0 0.01 18 10k 100 (100)

6.0 1.0 0.01 24 10k 100 (100)

6.0 1.0 0.01 32 10k 100 (100)

6.0 1.0 0.01 48 4k 40 (10)

6.0 1.0 0.01 64 4k 40(10)

5.7 1.0 0.01 12 10k 100(100)

5.7 1.0 0.01 18 10k 100(100)

5.7 1.0 0.01 24 10k 100(100)

5.7 1.0 0.01 32 10k 100(100)

Quenched Wilson action simulations with hypercubic lattices

Simulation parameters

Numerical result (1)

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=== Volume dependence at beta=6.0 ===

Flat in the IR region, but not suppressed.

Not diverge in the IR region.

All the data are on the same line.

For largest volume (64)4=(6.4fm) 4

Numerical result (2)

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=== Volume dependence at beta=5.7 ===

Flat in the IR region, but not suppressed.

Not diverge in the IR region.

All the data are on the same line.

For largest volume

(32)4=(5.4fm) 4

Numerical result (3)

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=== α-parameter dependence at beta=5.7 ===

In the UV region, small variation with α

In the IR region, large change with α?

For smallest α, we got better result.

Summary

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We try to calculate gluon propagators in the confinement region in the stochastic gauge fixing method with the Coulomb gauge.

For this new calculation, we need more information and arguments.

We find sign of an infrared suppression of gluon propagators.

Larger physical volume ?We find that the infrared gluons are strongly affected by variation of alpha-gauge parameter.

Why ? We need investigation of the lowest eigenvalue of FP operator, the relation of the sharp gauge, etc.

Method(5)

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=== Disadvantage for using SGF ===

Langevin step dependence

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Gauge fixing term

Gauge fixing term bab AADd

)(

α-paramter small, dτ small more computation time

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Numerical results of Gluon propagators

Volume dependence , beta dependence , alpha parameter dependence

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Numerical results (1)

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Numerical results (1)

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JPS２００６S 24

クーロンゲージ QCD　クーロンゲージ QCD におけるハミルトニアン

　クーロンゲージ QCD におけるファデーフポボフ

　グルーオン伝播関数の時間成分

3 2 2 3 31 1 ( ) ( , ) ( )2 2i iH d x E B d xd y x D x y y

3 21 1( , ) ( )( , ) ( , )zD x y d z

M x y M x y

2( )M gA

20 0( ) ( ) ( ) ( )g A x A y V x y P x y

24 4( ) ( , ) ( )V x y g D x y x y

瞬間力部分

遅延部分